Fixed point theorems for compact and nonexpansive mappings on starshaped domains (Preliminary communication)
In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if is a convex, -complete modular space satisfying the Fatou property and -uniformly convex for all , C a convex, -closed, -bounded subset of , a -nonexpansive mapping, then has a fixed point.
It is proved that: for every Banach space which has uniformly normal structure there exists a with the property: if is a nonempty bounded closed convex subset of and is an asymptotically regular mapping such that where is the Lipschitz constant (norm) of , then has a fixed point in .